ON SOLVABILITY AND ALGORITHM OF GENERAL STRONGLY NONLINEAR VARIATIONAL-LIKE INEQUALITIES

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 43, Issue 2, 2006, pp.319-331
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2006.43.2.319

Title & Authors

ON SOLVABILITY AND ALGORITHM OF GENERAL STRONGLY NONLINEAR VARIATIONAL-LIKE INEQUALITIES

Liu Zeqing; Sun, Juhe; Shim, Soo-Hak; Kang, Shin-Min;

Liu Zeqing; Sun, Juhe; Shim, Soo-Hak; Kang, Shin-Min;

Abstract

In this paper, a new class of general strongly nonlinear variational-like inequalities was introduced and studied. The existence and uniqueness of solutions and a new iterative algorithm for the general strongly nonlinear variational-like inequality are established and suggested, respectively. The convergence criteria of the iterative sequence generated by the iterative algorithm are also given.

Keywords

general strongly nonlinear variational-like inequality;existence and uniqueness;contracton mapping;

Language

English

References

1.

S. S. Chang, Variational inequalitity and complementarity theory with applica- tions, Shanghai Sci. Technol., Shanghai (1991)

2.

S. S. Chang, On the existence of solutions for a class of quasi-bilinear variational inequalities, J. Sys. Sci. Math. Scis. 16 (1996), 136-140 [In Chinese]

3.

P. Cubiotti, Existence of solutions for lower semicontinuous quasi-equilibrium problems, Comput. Math. Appl. 30 (1995), no. 12, 11-22

4.

X. P. Ding, Algorithm of solutions for mixed nonlinear variational-like inequal- ities in reflexive Banach space, Appl. Math. Mech. 19 (1998), no. 6, 521-529

5.

X. P. Ding, Existence and algorithm of solutions for nonlinear mixed variational- like inequalities in Banach spaces, J. Comput. Appl. Math. 157 (2003), no. 2, 419-434

6.

X. P. Ding and K. K. Tan, A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math. 63 (1992), no. 2, 233-247

7.

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North- Holland, Amsterdam, Holland, 1976

8.

N. J. Huang and C. X. Deng, Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities, J. Math. Anal. Appl. 256 (2001), no. 2, 345-359

9.

Z. Liu, J. S. Ume, and S. M. Kang, Nonlinear variational inequalities on reflexive Banach spaces and topological vector spaces, Int. J. Math. Math. Sci. 2003 (2003), no. 4, 199-207

10.

Z. Liu, L. Debnath, S. M. Kang, and J. S. Ume, Completely generalized multival- ued nonlinear quasi-variational inclusions, Int. J. Math. Math. Sci. 30 (2002), no. 10, 593-604

11.

Z. Liu, L. Debnath, S. M. Kang, and J. S. Ume, On the generalized nonlinear quasivariational inclusions, Acta. Math. Inform. Univ. Ostraviensis 11 (2003), no. 1, 81-90

12.

Z. Liu, L. Debnath, S. M. Kang, and J. S. Ume, Sensitivity analysis for parametric completely generalized nonlinear im- plicit quasivariational inclusions, J. Math. Anal. Appl. 277 (2003), no. 1, 142- 154

13.

Z. Liu, L. Debnath, S. M. Kang, and J. S. Ume, Generalized mixed quasivariational inclusions and generalized mixed resolvent equations for fuzzy mappings, Appl. Math. Comput. 149 (2004), no. 3, 879-891

14.

Z. Liu, S. M. Kang, and J. S. Ume, On general variational inclusions with noncompact valued mappings, Adv. Nonlinear Var. Inequal. 5 (2002), no. 2, 11-25

15.

Z. Liu, S. M. Kang, and J. S. Ume, Completely generalized multivalued strongly quasivariational inequali- ties, Publ. Math. Debrecen 62 (2003), no. 1-2, 187-204

16.

Z. Liu, S. M. Kang, and J. S. Ume, Generalized variational inclusions for fuzzy mappings, Adv. Nonlinear Var. Inequal. 6 (2003), no. 1, 31-40

17.

Z. Liu, S. M. Kang, and J. S. Ume, The solvability of a class of quasivariational inequalities, Adv. Nonlin- ear Var. Inequal. 6 (2003), no. 2, 69-78

18.

Z. Liu and S. M. Kang, Generalized multivalued nonlinear quasi-variational inclusions, Math. Nachr. 253 (2003), 45-54

19.

Z. Liu and S. M. Kang, Convergence and stability of perturbed three-step iterative algorithm for completely generalized nonlinear quasivariational inequalities, Appl. Math. Comput. 149 (2004), no. 1, 245-258

20.

Z. Liu, J. S. Ume, and S. M. Kang, General strongly nonlinear quasivariational inequalities with relaxed Lipschitz and relaxed monotone mappings, J. Optim. Theory Appl. 114 (2002), no. 3, 639-656

21.

Z. Liu, J. S. Ume, and S. M. Kang, Resolvent equations technique for general variational inclusions, Proc. Japan Acad., Ser. A Math. Sci. 78 (2002), no. 10, 188-193

22.

Z. Liu, J. S. Ume, and S. M. Kang, Nonlinear variational inequalities on reflexive Banach spaces and topo- logical vector spaces, Int. J. Math. Math. Sci. 2003 (2003), no. 4, 199-207

23.

Z. Liu, J. S. Ume, and S. M. Kang, Completely generalized quasivariational inequalities, Adv. Nonlinear Var. Inequal. 7 (2004), no. 1, 35-46

24.

P. D. Panagiotopoulos and G. E. Stavroulakis, New types of variational prin- ciples based on the notion of quasidifferentiability, Acta Mech. 94 (1992), no. 3-4, 171-194

25.

J. Parida and A. Sen, A variational-like inequality for multifunctions with ap- plications, J. Math. Anal. Appl. 124 (1987), no. 1, 73-81

26.

G. Tian, Generalized quasi-variational-like inequality problem, Math. Oper. Res. 18 (1993), no. 3, 752-764

27.

R. U. Verma, On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators, J. Math. Anal. Appl. 213 (1997), no. 1, 387- 392

28.

R. U. Verma, Generalized variational inequalities and associated nonlinear equations, Czechoslovak Math. J. 48 (1998), no. 3, 413-418

29.

R. U. Verma, Generalized pseudo-contractions and nonlinear variational inequalities, Publ. Math. Debrecen 53 (1998), no. 1-2, 23-28

30.

R. U. Verma, The solvability of a class of generalized nonlinear variational inequali- ties based on an iterative algorithm, Appl. Math. Lett. 12 (1999), no. 4, 51-53

31.

R. U. Verma, A general iterative algorithm and solvability of nonlinear quasivaria- tional inequalities, Adv. Nonlinear Var. Inequal. 4 (2001), no. 2, 79-87